Lesson 8: Composition of Linear Transformations
Transcript of Lesson 8: Composition of Linear Transformations
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
137
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Lesson 8: Composition of Linear Transformations
Student Outcomes
Students compose two linear transformations in the plane by multiplying the associated matrices.
Students visualize composition of linear transformations in the plane.
Lesson Notes
In this lesson, students examine the geometric effects of performing a sequence of two linear transformations in ℝ2
produced by various 2 × 2 matrices on the unit square. The GeoGebra demo TransformSquare (http://eureka-
math.org/G12M2L8/geogebra-TransformSquare) may be used in this lesson but is not necessary. This GeoGebra demo
allows students to input values in 2 × 2 matrices 𝐴 between −5 and 5 in increments of 0.1 and visually see the effect of
the transformation produced by the matrix 𝐴 on the unit square. If there are not enough computers for students to
access the demos in pairs or small groups, modify the lesson. Consider having students work in groups on
transformations. Then, as parts of the Exploratory Challenge are finished, show specific transformations to the entire
class using the teacher computer. Another option would be to have different groups come up and show the
transformation that they created by hand and then use the teacher computer to show that transformation using
software.
Classwork
Opening (3 minutes)
In this lesson, the transformation that arises from doing two linear transformations in sequence is considered.
Display each transformation matrix below on the board, and ask students what transformation the matrix represents.
[2 00 2
] A dilation with a scale factor of 2
[0 11 0
] A reflection across the line 𝑦 = 𝑥
[2 −11 2
] A rotation about the origin by 𝜃 = tan−1 (21) and a dilation with a scale factor of √22 + 12 = √5
How could we represent a composition of two of these transformations?
Opening Exercise (5 minutes)
The focus of this lesson is on students recognizing that a matrix produced by a composition of two linear transformations
in the plane is the product of the matrices of each transformation. Thus, students need to review the process of
multiplying 2 × 2 matrices. The products used in this Opening Exercise appear later in the lesson.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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Opening Exercise
Compute the product 𝑨𝑩 for the following pairs of matrices.
a. 𝑨 = [𝟎 −𝟏𝟏 𝟎
], 𝑩 = [𝟎 −𝟏𝟏 𝟎
]
𝑨𝑩 = [−𝟏 𝟎𝟎 −𝟏
]
b. 𝑨 = [𝟏 𝟎𝟎 −𝟏
], 𝑩 = [−𝟏 𝟎𝟎 𝟏
]
𝑨𝑩 = [−𝟏 𝟎𝟎 −𝟏
]
c. 𝑨 = [
√𝟐𝟐
−√𝟐𝟐
√𝟐𝟐
√𝟐𝟐
], 𝑩 = [𝟒 𝟎𝟎 𝟒
]
𝑨𝑩 = [𝟐√𝟐 −𝟐√𝟐
𝟐√𝟐 𝟐√𝟐]
d. 𝑨 = [𝟏 𝟏𝟎 𝟏
], 𝑩 = [𝟐 𝟎𝟎 𝟐
]
𝑨𝑩 = [𝟐 𝟐𝟎 𝟐
]
e. 𝑨 = [
𝟏𝟐
√𝟑𝟐
−√𝟑𝟐
𝟏𝟐
], 𝑩 = [
√𝟑𝟐
−𝟏𝟐
𝟏𝟐
√𝟑𝟐
]
𝑨𝑩 = [
√𝟑𝟐
𝟏𝟐
−𝟏𝟐
√𝟑𝟐
]
Exploratory Challenge (25 minutes)
For this challenge, students use the GeoGebra demo TransformSquare.ggb to transform a
unit square in the plane in order to explore the geometric effects of performing a
sequence of two linear transformations defined by matrix multiplication for different
matrices 𝐴 and 𝐵. While the software illustrates a single transformation via matrix
multiplication, students have to reason through the effects of doing two transformations
in sequence. If possible, allow students to access the software themselves. If there is no
way for students to access the software themselves, have them plot the transformed
vertices of the unit square and draw conclusions from there.
Scaffolding:
Provide struggling
students with colored
pencils or markers to use
to identify the three
figures. For example, use
blue for the original
square, green for the
square transformed by 𝐿𝐵,
and red for the square
transformed by 𝐿𝐵 and
then 𝐿𝐴.
Ask early finishers to figure
out whether or not the
geometric effect of
performing first 𝐿𝐴 and
then 𝐿𝐵 is the same as
performing first 𝐿𝐵 and
then 𝐿𝐴.
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PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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Exploratory Challenge
a. For each pair of matrices 𝑨 and 𝑩 given below:
1. Describe the geometric effect of the transformation 𝑳𝑩 ([𝒙𝒚]) = 𝑩 ⋅ [
𝒙𝒚].
2. Describe the geometric effect of the transformation 𝑳𝑨 ([𝒙𝒚]) = 𝑨 ⋅ [
𝒙𝒚].
3. Draw the image of the unit square under the transformation 𝑳𝑩 ([𝒙𝒚]) = 𝑩 ⋅ [
𝒙𝒚].
4. Draw the image of the transformed square under the transformation 𝑳𝑨 ([𝒙𝒚]) = 𝑨 ⋅ [
𝒙𝒚].
5. Describe the geometric effect on the unit square of performing first 𝑳𝑩 and then 𝑳𝑨.
6. Compute the matrix product 𝑨𝑩.
7. Describe the geometric effect of the transformation 𝑳𝑨𝑩 ([𝒙𝒚]) = 𝑨𝑩 ⋅ [
𝒙𝒚].
i. 𝑨 = [𝟎 −𝟏𝟏 𝟎
], 𝑩 = [𝟎 −𝟏𝟏 𝟎
]
1. The transformation produced by matrix 𝑩 has the effect of rotation by 𝟗𝟎° counterclockwise.
2. The transformation produced by matrix 𝑨 has the effect of rotation by 𝟗𝟎° counterclockwise.
3. The green square in the second quadrant is the image of the original square under the
transformation produced by 𝑩.
4. The red square in the third quadrant is the image of the square after transforming with matrix
𝑨 and then matrix 𝑩.
5. If we rotate twice by 𝟗𝟎°, then the net effect is a rotation by 𝟏𝟖𝟎°.
6. The matrix product is 𝑨𝑩 = [−𝟏 𝟎𝟎 −𝟏
].
7. The transformation produced by matrix 𝑨𝑩 has the effect of rotation by 𝟏𝟖𝟎°.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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ii. 𝑨 = [𝟏 𝟎𝟎 −𝟏
], 𝑩 = [−𝟏 𝟎𝟎 𝟏
]
1. The transformation produced by matrix 𝑩 has the effect of reflection across the 𝒚-axis.
2. The transformation produced by matrix 𝑨 has the effect of reflection across the 𝒙-axis.
3. The green square in the second quadrant is the image of the original square under the
transformation produced by 𝑩.
4. The red square in the third quadrant is the image of the square after transforming with matrix
𝑨 and then matrix 𝑩.
5. If we reflect across the 𝒚-axis and then reflect across the 𝒙-axis, the net effect is a rotation by
𝟏𝟖𝟎°.
6. The matrix product is 𝑨𝑩 = [−𝟏 𝟎𝟎 −𝟏
].
7. The transformation produced by matrix 𝑨𝑩 has the effect of rotation by 𝟏𝟖𝟎°.
iii. 𝑨 = [
√𝟐𝟐
−√𝟐𝟐
√𝟐𝟐
√𝟐𝟐
], 𝑩 = [𝟒 𝟎𝟎 𝟒
]
1. The transformation produced by matrix 𝑩 has the effect of dilation from the origin with scale
factor 𝟒.
2. The transformation produced by matrix 𝑨 has the effect of rotation by 𝟒𝟓°.
3. The large green square is the image of the original unit square under the transformation
produced by 𝑩.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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4. The tilted red square is the image of the green square under the transformation produced by 𝑨.
5. If we dilate with factor 𝟒 and then rotate by 𝟒𝟓°, the net effect is a rotation by 𝟒𝟓° and dilation
with scale factor 𝟒.
6. The matrix product is 𝑨𝑩 = [𝟐√𝟐 −𝟐√𝟐
𝟐√𝟐 𝟐√𝟐].
7. The transformation produced by matrix 𝑨𝑩 has the effect of rotation by 𝟒𝟓° while scaling by 𝟒.
iv. 𝑨 = [𝟏 𝟏𝟎 𝟏
], 𝑩 = [𝟐 𝟎𝟎 𝟐
]
1. The transformation produced by matrix 𝑩 has the effect of dilation with scale factor 𝟐.
2. The transformation produced by matrix 𝑨 has the effect of shearing parallel to the 𝒚-axis.
3. The large green square is the image of the original unit square under the transformation
produced by 𝑩.
4. The red parallelogram is the image of the green square under the transformation produced by 𝑨.
5. If we dilate and then shear, the net effect is a dilated shear.
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PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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6. The matrix product is 𝑨𝑩 = [𝟐 𝟐𝟎 𝟐
].
7. The transformation produced by matrix 𝑨𝑩 has the effect of a dilated shear.
v. 𝑨 = [
𝟏𝟐
√𝟑𝟐
−√𝟑𝟐
𝟏𝟐
], 𝑩 = [
√𝟑𝟐
−𝟏𝟐
𝟏𝟐
√𝟑𝟐
]
1. The transformation produced by matrix 𝑩 has the effect of rotation by 𝟑𝟎°.
2. The transformation produced by matrix 𝑨 has the effect of rotation by −𝟔𝟎°.
3. The tilted green square is the image of the original unit square under the transformation
produced by 𝑩.
4. The tilted red square is the image of the green square under the transformation produced by 𝑨.
5. If we rotate by 𝟑𝟎° counterclockwise and then rotate by 𝟔𝟎° clockwise, the net result is a rotation
by 𝟑𝟎° counterclockwise.
6. The matrix product is 𝑨𝑩 = [
√𝟑𝟐
𝟏𝟐
−𝟏𝟐
√𝟑𝟐
].
7. The transformation produced by matrix 𝑨𝑩 has the effect of rotation by −𝟑𝟎°.
b. Make a conjecture about the geometric effect of the linear transformation produced by the matrix 𝑨𝑩.
Justify your answer.
The linear transformation produced by matrix 𝑨𝑩 has the same geometric effect of the sequence of the linear
transformation produced by matrix 𝑩 followed by the linear transformation produced by matrix 𝑨. We have
seen in previous work that if we multiply by 𝑨𝑩, we get the same transformation as when we multiplied by 𝑨
first and then multiplied the result by 𝑩.
MP.3 &
MP.2
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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Lesson Summary
The linear transformation produced by matrix 𝑨𝑩 has the same geometric effect as the sequence of the linear
transformation produced by matrix 𝑩 followed by the linear transformation produced by matrix 𝑨.
That is, if matrices 𝑨 and 𝑩 produce linear transformations 𝑳𝑨 and 𝑳𝑩 in the plane, respectively, then the linear
transformation 𝑳𝑨𝑩 produced by the matrix 𝑨𝑩 satisfies
𝑳𝑨𝑩 ([𝒙𝒚]) = 𝑳𝑨 (𝑳𝑩 ([
𝒙𝒚])).
Discussion (5 minutes)
This Discussion justifies why the conjecture students made at the end of Exploratory Challenge is valid.
What was the conjecture you made at the end of the Exploratory Challenge?
The transformation that comes from the matrix 𝐴𝐵 has the same geometric effect as the sequence of
transformations that come from the matrix 𝐵 first and then 𝐴.
Why does matrix 𝐴𝐵 represent 𝐵 first and then 𝐴? Doesn’t that seem backward?
If we look at this as multiplication, we are multiplying
𝐿𝐴𝐵 ([𝑥𝑦]) = (𝐴𝐵) ⋅ [
𝑥𝑦]
= 𝐴(𝐵 ⋅ [𝑥𝑦])
= 𝐿𝐴 (𝐿𝐵 ([𝑥𝑦])) ,
so the point (𝑥, 𝑦) is transformed first by 𝐿𝐵, and then the result is transformed by 𝐿𝐴. It looks backward, but
the transformation on the right happens to our point first. We saw this phenomenon in Geometry when we
represented a rotation by 𝜃 about the origin by 𝑅0,𝜃 and a reflection across line ℓ by 𝑟ℓ. Then, doing the
rotation and then the reflection is denoted by 𝑟ℓ ∘ 𝑅𝑂,𝜃, which represents the combined effect of first
performing the rotation 𝑅𝑂,𝜃 and then the reflection 𝑟ℓ.
Closing (3 minutes)
Ask students to summarize the key points of the lesson in writing or to a partner. Some important summary elements
are listed below.
Exit Ticket (4 minutes)
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PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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Name Date
Lesson 8: Composition of Linear Transformations
Exit Ticket
Let 𝐴 be the matrix representing a rotation about the origin 135° and 𝐵 be the matrix representing a dilation of 6. Let
𝑥 = [−112
].
a. Write down 𝐴 and 𝐵.
b. Find the matrix representing a dilation of 𝑥 by 6, followed by a rotation about the origin of 135°.
c. Graph and label 𝑥, 𝑥 after a dilation of 6, and 𝑥 after both transformations have been applied.
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PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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Exit Ticket Sample Solutions
Let 𝑨 be the matrix representing a rotation about the origin 𝟏𝟑𝟓° and 𝑩 be the matrix representing a dilation of 𝟔. Let
𝒙 = [−𝟏𝟏𝟐
].
a. Write down 𝑨 and 𝑩.
𝑨 =
[ −
√𝟐
𝟐−
√𝟐
𝟐
√𝟐
𝟐−
√𝟐
𝟐 ]
, 𝑩 = [𝟔 𝟎𝟎 𝟔
]
b. Find the matrix representing a dilation 𝒙 by 𝟔, followed by a rotation about the origin of 𝟏𝟑𝟓°.
𝑨𝑩 = [−𝟑√𝟐 −𝟑√𝟐
𝟑√𝟐 −𝟑√𝟐]
c. Graph and label 𝒙, 𝒙 after a dilation of 𝟔, and 𝒙 after both transformations have been applied.
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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Problem Set Sample Solutions
1. Let 𝑨 be the matrix representing a dilation of 𝟏
𝟐, and let 𝑩 be the matrix representing a reflection across the 𝒚-axis.
a. Write 𝑨 and 𝑩.
𝑨 = [
𝟏
𝟐𝟎
𝟎𝟏
𝟐
] , 𝑩 = [−𝟏 𝟎𝟎 𝟏
]
b. Evaluate 𝑨𝑩. What does this matrix represent?
𝑨𝑩 = [−
𝟏
𝟐𝟎
𝟎𝟏
𝟐
]
𝑨𝑩 is a reflection across the 𝒚-axis followed by a dilation of 𝟏
𝟐.
c. Let 𝒙 = [𝟓𝟔], 𝒚 = [
−𝟏𝟑
], and 𝒛 = [𝟖
−𝟐]. Find (𝑨𝑩)𝒙, (𝑨𝑩)𝒚, and (𝑨𝑩)𝒛.
(𝑨𝑩)𝒙 = [−𝟓
𝟐𝟑
] , (𝑨𝑩)𝒚 = [
𝟏
𝟐𝟑
𝟐
] , (𝑨𝑩)𝒛 = [−𝟒−𝟏
]
2. Let 𝑨 be the matrix representing a rotation of 𝟑𝟎°, and let 𝑩 be the matrix representing a dilation of 𝟓.
a. Write 𝑨 and 𝑩.
𝑨 =
[ √𝟑
𝟐−
𝟏
𝟐𝟏
𝟐
√𝟑
𝟐 ]
, 𝑩 = [𝟓 𝟎𝟎 𝟓
]
b. Evaluate 𝑨𝑩. What does this matrix represent?
𝑨𝑩 =
[ 𝟓√𝟑
𝟐−
𝟓
𝟐𝟓
𝟐
𝟓√𝟑
𝟐 ]
𝑨𝑩 is a dilation of 𝟓 followed by a rotation of 𝟑𝟎°.
c. Let 𝒙 = [𝟏𝟎]. Find (𝑨𝑩)𝒙.
(𝑨𝑩)𝒙 =
[ 𝟓√𝟑
𝟐𝟓
𝟐 ]
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PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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3. Let 𝑨 be the matrix representing a dilation of 𝟑, and let 𝑩 be the matrix representing a reflection across the line
𝒚 = 𝒙.
a. Write 𝑨 and 𝑩.
𝑨 = [𝟑 𝟎𝟎 𝟑
] , 𝑩 = [𝟎 𝟏𝟏 𝟎
]
b. Evaluate 𝑨𝑩. What does this matrix represent?
𝑨𝑩 = [𝟎 𝟑𝟑 𝟎
]
𝑨𝑩 is a reflection across the line 𝒚 = 𝒙 followed by a dilation of 𝟑.
c. Let 𝒙 = [−𝟐𝟕
]. Find (𝑨𝑩)𝒙.
(𝑨𝑩)𝒙 = [𝟐𝟏−𝟔
]
4. Let 𝑨 = [𝟑 𝟎𝟑 𝟑
] 𝐚𝐧𝐝 𝑩 = [𝟎 −𝟏𝟏 𝟎
].
a. Evaluate 𝑨𝑩.
[𝟎 −𝟑𝟑 −𝟑
]
b. Let 𝒙 = [−𝟐𝟐
]. Find (𝑨𝑩)𝒙.
[−𝟔−𝟏𝟐
]
c. Graph 𝒙 and (𝑨𝑩)𝒙.
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PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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5. Let 𝑨 = [
𝟏𝟑
𝟎
𝟐𝟏𝟑
] and 𝑩 = [𝟑 𝟏𝟏 −𝟑
].
a. Evaluate 𝑨𝑩.
[𝟏
𝟏
𝟑𝟏𝟗
𝟑𝟏
]
b. Let 𝒙 = [𝟎𝟑]. Find (𝑨𝑩)𝒙.
[𝟏𝟑]
c. Graph 𝒙 and (𝑨𝑩)𝒙.
6. Let 𝑨 = [𝟐 𝟐𝟐 𝟎
] and 𝑩 = [𝟎 𝟐𝟐 𝟐
].
a. Evaluate 𝑨𝑩.
[𝟒 𝟖𝟎 𝟒
]
b. Let 𝒙 = [𝟑
−𝟐]. Find (𝑨𝑩)𝒙.
[−𝟒−𝟖
]
NYS COMMON CORE MATHEMATICS CURRICULUM M2 Lesson 8
PRECALCULUS AND ADVANCED TOPICS
Lesson 8: Composition of Linear Transformations
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c. Graph 𝒙 and (𝑨𝑩)𝒙.
7. Let 𝑨, 𝑩, 𝑪 be 𝟐 × 𝟐 matrices representing linear transformations.
a. What does 𝑨(𝑩𝑪) represent?
𝑨(𝑩𝑪) represents the linear transformation of applying the transformation that 𝑪 represents, followed by the
transformation that 𝑩 represents, followed by the transformation that 𝑨 represents.
b. Will the pattern established in part (a) be true no matter how many matrices are multiplied on the left?
Yes, in general. When you multiply by a matrix on the left, you are applying a linear transformation after all
linear transformations to the right have been applied.
c. Does (𝑨𝑩)𝑪 represent something different from 𝑨(𝑩𝑪)? Explain.
No, it does not. This is the linear transformation obtained by applying 𝑪 and then 𝑨𝑩, which is 𝑩 followed by
𝑨.
8. Let 𝑨𝑩 represent any composition of linear transformations in ℝ𝟐. What is the value of (𝑨𝑩)𝒙 where 𝒙 = [𝟎𝟎]?
Since a composition of linear transformations in ℝ𝟐 is also a linear transformation, we know that applying it to the
origin will result in no change.
MP.7